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<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">o</font> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">f</font>] &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">o</font> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> <br/> for <font color="Olive" title="b3">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) ) ) )</span><br/>

<b>take </b><font color="Maroon" title="c7">f</font> =  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font>] &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">o</font> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> <br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><br/>

<b>thus </b><a NAME="E1:53_6"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c7">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">o</font> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> <br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c8">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">o</font> <a href="cat_5.html#K13" title="CAT_5:func.13">Hom</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Olive" title="b1">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">g</font>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">f</font>,<font color="Olive" title="b1">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c9">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) )</span><br/>



<b>thus </b><a NAME="E2:53_6"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c9">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> implies <font color="Maroon" title="c9">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> )</span><br/>

<b>assume </b><a NAME="E3:53_6"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c9">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font></span><br/>

<a NAME="E4:53_6"/><b>then </b>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/>
<b>hence </b><a NAME="E5:53_6"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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